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Attractors for model of polymer solutions motion
Weak solvability of fractional Voigt model of viscoelasticity
Laboratory of Mathematical Fluid Dynamics, Voronezh State University, Universitetskaya pl., 1, Voronezh 394 018, Russia |
In the present paper we establish the existence of weak solutions to one fractional Voigt type model of viscoelastic fluid. This model takes into account a memory along the motion trajectories. The investigation is based on the theory of regular Lagrangean flows, approximation of the problem under consideration by a sequence of regularized Navier-Stokes systems and the following passage to the limit.
References:
| [1] |
L. Ambrosio,
Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260.
doi: 10.1007/s00222-004-0367-2. |
| [2] |
M. Caputo and F. Mainardi,
Linear models of dissipation in inelastic solidss, La Rivista del Nuovo Cimento, 1 (1971), 161-198.
doi: 10.1007/BF02820620. |
| [3] |
M. Caputo and F. Mainardi,
A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147.
doi: 10.1007/BF00879562. |
| [4] |
D. Cordoba, C. Fefferman and R. de la Llave,
On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36 (2000), 204-213.
doi: 10.1137/S0036141003424095. |
| [5] |
G. Crippa and C. de Lellis,
The regularity results for diPernaions flow, J. Reine Angew. Math., 616 (2008), 15-46.
doi: 10.1515/CRELLE.2008.016. |
| [6] |
G. Crippa,
Ordinary differential equations with non-Lipschitz vector fields, Boll. Union Mat. Ital., 1 (2008), 333-348.
|
| [7] |
R. J. DiPerna and P. L. Lions,
Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
| [8] |
A. N. Gerasimov,
A generalization of linear laws of deformation and its application to objectives of internal friction, Prikl. Mat. Mech., 12 (1948), 251-260.
|
| [9] |
I. Gyarmati,
Non-equilibrium Thermodynamics, Field Theory and Variational Principles, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
doi: 10.1007/978-3-642-51067-0. |
| [10] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo,
Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Vol. 204, Elsevier, Amsterdam, 2006. |
| [11] |
R. de la Llave and R. Obaya,
Regularity of the composition operator in spaces of H$\rmö$lder functions, Discrete Contin. Dyn. Syst. Ser. A, 5 (1999), 157-184.
|
| [12] |
F. Mainardi and G. Spada,
Creep, relaxation and viscosity for basic fractional models in rheology, Eur. Phys. J. Spec. Top., 193 (2011), 133-160.
doi: 10.1140/epjst/e2011-01387-1. |
| [13] |
E. N. Ogorodnikov, V. P. Radchenko and N. S. Yashagin,
Rheological model the viscoelastic body with memory and differential equations of fractional oscillators, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 22 (2011), 255-268.
|
| [14] |
V. P. Orlov and P. E. Sobolevskii,
On mathematical model of viscoelasticity with a memory, Differ. Integral Equ., 4 (1991), 103-115.
|
| [15] |
A. P. Oskolkov,
On some quasilinears systems occuring in studing of motion of viscous fluids, Zap. Nauchn. Sem. LOMI, 52 (1975), 128-157.
|
| [16] |
S. G. Samko, A. A. Kilbas and O. I. Marichev,
Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka I Tekhnika, Minsk, 1987. |
| [17] |
G. Scott-Blair,
Survey of General and Applied Rheology, 2$^{nd}$ edition, Isaac Pitman and Sons, London, 1949. |
| [18] |
P. E. Sobolewskii,
On equations of parabolic type in a Banach space, Trans. Moscow Math. Soc., 10 (1961), 297-350.
|
| [19] |
R. Temam,
Navier-Stokes Equation, North Holland Publishing Company, Amsterdam-New York-Oxford, 1979. |
| [20] |
V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of regularized model of a viscoelastic fluid, (Russian)Dokl. Akad. Nauk, 380 (2001), 308-311.
doi: 10.1023/A:1023860129831. |
| [21] |
V. G. Zvyagin and V. T. Dmitrienko,
On weak solutions of an initial-boundary value problem for the equation of motion of a viscoelastic fluid, Dokl. Math., 64 (2001), 190-193.
|
| [22] |
V. G. Zvyagin and M. V. Turbin,
The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci., 168 (2010), 157-308.
doi: 10.1007/s10958-010-9981-2. |
| [23] |
V. G. Zvyagin and D. A. Vorotnikov,
Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin-New York, 2008.
doi: 10.1515/9783110208283. |
show all references
References:
| [1] |
L. Ambrosio,
Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260.
doi: 10.1007/s00222-004-0367-2. |
| [2] |
M. Caputo and F. Mainardi,
Linear models of dissipation in inelastic solidss, La Rivista del Nuovo Cimento, 1 (1971), 161-198.
doi: 10.1007/BF02820620. |
| [3] |
M. Caputo and F. Mainardi,
A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147.
doi: 10.1007/BF00879562. |
| [4] |
D. Cordoba, C. Fefferman and R. de la Llave,
On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36 (2000), 204-213.
doi: 10.1137/S0036141003424095. |
| [5] |
G. Crippa and C. de Lellis,
The regularity results for diPernaions flow, J. Reine Angew. Math., 616 (2008), 15-46.
doi: 10.1515/CRELLE.2008.016. |
| [6] |
G. Crippa,
Ordinary differential equations with non-Lipschitz vector fields, Boll. Union Mat. Ital., 1 (2008), 333-348.
|
| [7] |
R. J. DiPerna and P. L. Lions,
Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
| [8] |
A. N. Gerasimov,
A generalization of linear laws of deformation and its application to objectives of internal friction, Prikl. Mat. Mech., 12 (1948), 251-260.
|
| [9] |
I. Gyarmati,
Non-equilibrium Thermodynamics, Field Theory and Variational Principles, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
doi: 10.1007/978-3-642-51067-0. |
| [10] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo,
Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Vol. 204, Elsevier, Amsterdam, 2006. |
| [11] |
R. de la Llave and R. Obaya,
Regularity of the composition operator in spaces of H$\rmö$lder functions, Discrete Contin. Dyn. Syst. Ser. A, 5 (1999), 157-184.
|
| [12] |
F. Mainardi and G. Spada,
Creep, relaxation and viscosity for basic fractional models in rheology, Eur. Phys. J. Spec. Top., 193 (2011), 133-160.
doi: 10.1140/epjst/e2011-01387-1. |
| [13] |
E. N. Ogorodnikov, V. P. Radchenko and N. S. Yashagin,
Rheological model the viscoelastic body with memory and differential equations of fractional oscillators, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 22 (2011), 255-268.
|
| [14] |
V. P. Orlov and P. E. Sobolevskii,
On mathematical model of viscoelasticity with a memory, Differ. Integral Equ., 4 (1991), 103-115.
|
| [15] |
A. P. Oskolkov,
On some quasilinears systems occuring in studing of motion of viscous fluids, Zap. Nauchn. Sem. LOMI, 52 (1975), 128-157.
|
| [16] |
S. G. Samko, A. A. Kilbas and O. I. Marichev,
Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka I Tekhnika, Minsk, 1987. |
| [17] |
G. Scott-Blair,
Survey of General and Applied Rheology, 2$^{nd}$ edition, Isaac Pitman and Sons, London, 1949. |
| [18] |
P. E. Sobolewskii,
On equations of parabolic type in a Banach space, Trans. Moscow Math. Soc., 10 (1961), 297-350.
|
| [19] |
R. Temam,
Navier-Stokes Equation, North Holland Publishing Company, Amsterdam-New York-Oxford, 1979. |
| [20] |
V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of regularized model of a viscoelastic fluid, (Russian)Dokl. Akad. Nauk, 380 (2001), 308-311.
doi: 10.1023/A:1023860129831. |
| [21] |
V. G. Zvyagin and V. T. Dmitrienko,
On weak solutions of an initial-boundary value problem for the equation of motion of a viscoelastic fluid, Dokl. Math., 64 (2001), 190-193.
|
| [22] |
V. G. Zvyagin and M. V. Turbin,
The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci., 168 (2010), 157-308.
doi: 10.1007/s10958-010-9981-2. |
| [23] |
V. G. Zvyagin and D. A. Vorotnikov,
Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin-New York, 2008.
doi: 10.1515/9783110208283. |
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